CN102384746B - Chebyshev output method for space motion state of rigid body - Google Patents

Abstract

The invention discloses a Chebyshev output method for the space motion state of a rigid body. The method comprises the steps of: defining a ternary number so that three speed components of the set of axes of the body and the ternary number form a system of linear differential equations; and performing approximation description on rolling, pitching and yawing angular speeds p, q and r by using Shifted Chebyshev orthogonal polynomials, wherein the state transfer matrix of the system can be solved in the manner of any order holder, thus obtaining the expression of the discrete state equation of the motion of the rigid body; and the problem of singularity of the attitude equation is avoided; therefore, the main motion state of the rigid body is obtained. The Chebyshev output method is characterized in that the ternary number is introduced so that the state transfer matrix is in the form of portioned upper triangles and thereby can be solved through deflation; therefore, the complexity of calculation is greatly simplified; and the Chebyshev output method is convenient for engineering use.

Description

A kind of modeling method of Chebyshev's output of rigid space motion state

Technical field

The present invention relates to spatial movement rigid model, particularly the large maneuvering flight State-output of aircraft modeling problem.

Background technology

Axis is that the rigid motion differential equation is the fundamental equation of describing the spatial movements such as aircraft, torpedo, spacecraft.Conventionally, in the application such as data processing, the state variable of axon system mainly comprises the X of 3 speed components, three Eulerian angle and earth axes _e, Y _e, Z _edeng, due to Z _ebe defined as vertical ground and point to ground ball center, therefore Z _ereality is negative flying height; X _e, Y _econventionally main GPS, GNSS, the Big Dipper etc. of relying on directly provide; Eulerian angle represent rigid space motion attitude, and the differential equation of portraying rigid body attitude is core wherein, is that pitching, rolling and crab angle are described conventionally with three Eulerian angle.When the angle of pitch of rigid body is+90 °, roll angle and crab angle cannot definite values, and it is excessive that the region of simultaneously closing on this singular point solves error, causes intolerable error in engineering and can not use; For fear of this problem, first people adopt the method for restriction angle of pitch span, and this degenerates equation, attitude work entirely, thereby be difficult to be widely used in engineering practice.Along with the research to aircraft extreme flight, people have adopted again direction cosine method, Rotation Vector, Quaternion Method etc. to calculate rigid motion attitude in succession.

Direction cosine method has been avoided " unusual " phenomenon of Eulerian angle describing methods, and with direction cosine method, calculating attitude matrix does not have equation degenerate problem, attitude work entirely, but need to solve 9 differential equations, calculated amount is larger, and real-time is poor, cannot meet engineering practice requirement.Rotation Vector is as list sample recursion, Shuangzi sample gyration vector, three increment gyration vectors and four increment rotating vector methods and various correction algorithms on this basis and recursive algorithm etc.While studying rotating vector in document, it is all the algorithm that is output as angle increment based on rate gyro.But in Practical Project, the output of some gyros is angle rate signals, as optical fibre gyro, dynamic tuned gyroscope etc.When rate gyro is output as angle rate signal, the Algorithm Error of rotating vector method obviously increases.Quaternion Method is that the function of 4 Eulerian angle of definition calculates boat appearance, can effectively make up the singularity of Eulerian angle describing method, as long as separate 4 differential equation of first order formula groups, than direction cosine attitude matrix differential equation calculated amount, there is obvious minimizing, can meet the requirement to real-time in engineering practice.Its conventional computing method have the card of finishing approximatioss, second order, fourth-order Runge-Kutta method and three rank Taylor expansions etc.Finishing card approximatioss essence is list sample algorithm, and what limited rotation was caused can not compensate by exchange error, and the algorithm drift under high current intelligence in attitude algorithm can be very serious.Adopt fourth-order Runge-Kutta method while solving quaternion differential equation, along with the continuous accumulation of integral error, there will be exceed ± 1 phenomenon of trigonometric function value, thereby cause calculating, disperse; Taylor expansion is also because the deficiency of computational accuracy is restricted.When rigid body is large when motor-driven, angular speed causes more greatly the error of said method larger; Moreover, the error that attitude is estimated usually can cause the error of 4 components of speed, highly output sharply to increase.

Summary of the invention

In order to overcome the large problem of existing rigid motion model output error, the invention provides a kind of modeling method of Chebyshev's output of rigid space motion state, the method is by definition Three-ary Number, making axis is that three speed components and Three-ary Number form linear differential equation group, and adopt Shifted Chebyshev (variation Chebyshev) orthogonal polynomial to rolling, pitching, yaw rate p, q, r carries out close approximation description, can be according to the state-transition matrix of the mode solving system of arbitrary order retainer, and then obtain the expression formula of rigid motion discrete state equations, avoided attitude equation singular problem, thereby obtain rigid body main movement state.

The present invention solves the technical scheme that its technical matters adopts, a kind of modeling method of Chebyshev's output of rigid space motion state, and its feature comprises the following steps:

1, axis is that three speed components are output as:

$\begin{array}{c}{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}+{\mathrm{g\Φ}}_v[(k+1)T,\mathrm{kT}]{\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}\\ +g{\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Φ}}_v[(k+1)T,\mathrm{\τ}]\left[\begin{array}{c}{n}_x\\ n_y\\ n_z\end{array}\right]\mathrm{d\τ}\end{array}$

Wherein: u, V, it is x that w is respectively along rigid body axis, y, the speed component of z axle, n _x, n _y, n _zbe respectively along x, y, the overload of z axle, g is acceleration of gravity, s ₁, s ₂, S ₃for the Three-ary Number of definition, and

${\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}{]\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}$

${\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_v\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_v^T-{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_v\mathrm{H\ξ}\left(\mathrm{kT}\right)$

${\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_s\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_s^T-{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_s\mathrm{H\ξ}\left(\mathrm{kT}\right)$

P, q, r is respectively rolling, pitching, yaw rate, and T is the sampling period; Parameter-definition is identical in full;

$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathrm{\ξ}\left(t\right)={\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T,$

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-2t/b\\ {\mathrm{\ξ}}_2\left(t\right)=8{(t/b)}^2-8(t/b)+1\\ .\\ .\\ .\\ {\mathrm{\ξ}}_{i+1}\left(t\right)={2\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},...,n-\mathrm{1,0}\≤t\≤\mathrm{NT}$

For the recursive form of Chebyshev (Chebyshev) orthogonal polynomial, b=NT, rolling, pitching, yaw rate p, q, the expansion of r is respectively

$p\left(t\right)=\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$q\left(t\right)=\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$r\left(t\right)=\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$\begin{array}{c}{\mathrm{\Π}}_v=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& -1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

$\begin{array}{c}{\mathrm{\Π}}_s=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

Work as p, q, when the high-order term n of expansion of r is odd number, m=4,6 ..., n+1, m=5 when high-order term n is even number, 7 ..., n+1, H _i(i=1,2 ..., n) be the corresponding row vector of H,

2, be highly output as:

$\stackrel{\·}{h}=\left[\begin{array}{ccc}u& v& w\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$

Wherein: h is height;

3, attitude angle is output as:

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t\frac{q{s}_2\left(t\right)+{\mathrm{rs}}_3\left(t\right)}{s_2^2\left(t\right)+s_3^2\left(t\right)}\mathrm{dt}$

Wherein: represent respectively rolling, pitching, crab angle, $\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]={\mathrm{\Φ}}_s(t,\mathrm{kT}){\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}.}$

The invention has the beneficial effects as follows: by introducing Three-ary Number, to make state-transition matrix be triangular form on piecemeal, can depression of order solving state transition matrix, greatly simplified computation complexity, be convenient to engineering and use.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

1, axis is that three speed components are output as:

$\begin{array}{c}{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}+{\mathrm{g\Φ}}_v[(k+1)T,\mathrm{kT}]{\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}\\ +g{\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Φ}}_v[(k+1)T,\mathrm{\τ}]\left[\begin{array}{c}{n}_x\\ n_y\\ n_z\end{array}\right]\mathrm{d\τ}\end{array}$

Wherein: u, V, it is x that w is respectively along rigid body axis, y, the speed component of z axle, n _x, n _y, n _zbe respectively along x, y, the overload of z axle, g is acceleration of gravity, s ₁, s ₂, s ₃for the Three-ary Number of definition, and

${\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}{]\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}$

${\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_v\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_v^T-{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_v\mathrm{H\ξ}\left(\mathrm{kT}\right)$

${\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_s\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_s^T-{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_s\mathrm{H\ξ}\left(\mathrm{kT}\right)$

P, q, r is respectively rolling, pitching, yaw rate, and T is the sampling period; Parameter-definition is identical in full;

$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathrm{\ξ}\left(t\right)={\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T,$

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-2t/b\\ {\mathrm{\ξ}}_2\left(t\right)=8{(t/b)}^2-8(t/b)+1\\ .\\ .\\ .\\ {\mathrm{\ξ}}_{i+1}\left(t\right)={2\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},...,n-\mathrm{1,0}\≤t\≤\mathrm{NT}$

For the recursive form of Chebyshev (Chebyshev) orthogonal polynomial, b=NT, rolling, pitching, yaw rate p, q, the expansion of r is respectively

$p\left(t\right)=\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$q\left(t\right)=\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$r\left(t\right)=\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$\begin{array}{c}{\mathrm{\Π}}_v=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& -1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

$\begin{array}{c}{\mathrm{\Π}}_s=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

Work as p, q, when the high-order term n of expansion of r is odd number, m=4,6 ..., n+1, m=5 when high-order term n is even number, 7 ..., n+1, H _i(i=1,2 ..., n) be the corresponding row vector of H,

2, be highly output as:

$\stackrel{\·}{h}=\left[\begin{array}{ccc}u& v& w\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$

Wherein: h is height;

3, attitude angle is output as:.

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t\frac{q{s}_2\left(t\right)+{\mathrm{rs}}_3\left(t\right)}{s_2^2\left(t\right)+s_3^2\left(t\right)}\mathrm{dt}$

Wherein:

represent respectively rolling, pitching, crab angle, $\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]={\mathrm{\Φ}}_s(t,\mathrm{kT}){\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}.}$

Claims (1)

1. a modeling method for the Chebyshev of rigid space motion state output, its feature comprises the following steps: axis is that three speed components are output as:

$\begin{array}{c}{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}u\left(t\right)\\ v\left(t\right)\\ w\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}+{\mathrm{g\Φ}}_v[(k+1)T,\mathrm{kT}]{\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]{\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}\\ +g{\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Φ}}_v[(k+1)T,\mathrm{\τ}]\left[\begin{array}{c}{n}_x\\ n_y\\ n_z\end{array}\right]\mathrm{d\τ}\end{array}$

Wherein: u, v, it is x that w is respectively along rigid body axis, y, the speed component of z axle, n _x, n _y, n _zbe respectively along x, y, the overload of z axle, g is acceleration of gravity, s ₁, s ₂, s ₃for the Three-ary Number of definition, and

${\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=(k+1)T}={\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}{]\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}}$

${\mathrm{\Φ}}_v[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_v\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_v^T-{\mathrm{\Π}}_v\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_v\mathrm{H\ξ}\left(\mathrm{kT}\right)$

${\mathrm{\Φ}}_s[(k+1)T,\mathrm{kT}]\≈I+{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+{\mathrm{\Π}}_s\{P\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{\mathrm{\Π}}_s^T-{\mathrm{\Π}}_s\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\Π}}_s\mathrm{H\ξ}\left(\mathrm{kT}\right)$

P, q, r is respectively rolling, pitching, yaw rate, and T is the sampling period;

$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathrm{\ξ}\left(t\right)={\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T,$

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-2t/b\\ {\mathrm{\ξ}}_2\left(t\right)=8{(t/b)}^2-8(t/b)+1\\ .\\ .\\ .\\ {\mathrm{\ξ}}_{i+1}\left(t\right)={2\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},...,n-\mathrm{1,0}\≤t\≤\mathrm{NT}$

For the recursive form of Chebyshev's orthogonal polynomial, b=NT, rolling, pitching, yaw rate p, q, the expansion of r is respectively

$p\left(t\right)=\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$q\left(t\right)=\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$r\left(t\right)=\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]{\left[\begin{array}{ccccc}{\mathrm{\ξ}}_0\left(t\right)& {\mathrm{\ξ}}_1\left(t\right)& ...& {\mathrm{\ξ}}_{n-1}\left(t\right)& {\mathrm{\ξ}}_n\left(t\right)\end{array}\right]}^T$

$\begin{array}{c}{\mathrm{\Π}}_v=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& -1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

$\begin{array}{c}{\mathrm{\Π}}_s=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& ...& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& ...& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& ...& r_{n-1}& r_n\end{array}\right]\end{array}$

Work as p, q, when the high-order term n of expansion of r is odd number, m=4,6 ..., n+1, m=5 when high-order term n is even number, 7 ..., n+1, H _i(i=1,2 ..., n) be the corresponding row vector of H,

Highly be output as:

$\stackrel{\·}{h}=\left[\begin{array}{ccc}u& v& w\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$

Wherein: h is height;

Attitude angle is output as:

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t\frac{q{s}_2\left(t\right)+{\mathrm{rs}}_3\left(t\right)}{s_2^2\left(t\right)+s_3^2\left(t\right)}\mathrm{dt}$

Wherein:

represent respectively rolling, pitching, crab angle, $\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]={\mathrm{\Φ}}_s(t,\mathrm{kT}){\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\\ s_3\left(t\right)\end{array}\right]}_{t=\mathrm{kT}.}$